Linear Algebra · Lesson 01

What Is a Vector?

See how an arrow, a displacement, and a pair of coordinates describe the same idea.

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v = (x y)

∥ v ∥ = x^{2} + y^{2}

P Q = Q - P

R^{2}

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An instruction to move.

Imagine giving someone directions on a blank map: move three units to the right, then two units up. You have not named a place. You have described a movement. A vector captures that movement as a single object.

Draw an arrow from where you start to where you finish. Its direction tells you which way to move; its length tells you how far. The two-step route is useful for measuring the movement, but the vector itself points directly from start to finish.

Read the cyan arrow as one movement. The dashed legs separate its horizontal and vertical components; the gold point marks the endpoint.

Give the arrow coordinates.

Choose perpendicular horizontal and vertical axes with the same unit scale. The first component measures the horizontal displacement; the second measures the vertical displacement. Positive means right or up, negative means left or down.

\mathbf{v}=\begin{bmatrix}3\\2\end{bmatrix}

The column is a compact description of the arrow, not a second kind of object. Read it from top to bottom: horizontal, then vertical. These components depend on the coordinate system we chose; a different basis can give the same geometric vector different coordinates.

Components are not the length.

The horizontal and vertical legs form a right triangle. The arrow is its hypotenuse, so the Pythagorean theorem gives its magnitude. Adding the components would measure the two-leg route instead of the straight-line displacement.

\|\mathbf{v}\|=\sqrt{13}\approx 3.61

A new starting point. The same vector.

Now start away from the origin. To find the displacement, subtract the starting coordinates from the ending coordinates. The vector records the change in position, not the endpoint by itself.

The arrow has moved, but its components and length have not. A point describes a location; a vector describes the displacement between two locations.

Worked example

From P to Q

Subtract start from finish, one coordinate at a time.

\overrightarrow{PQ}=\begin{bmatrix}4\\3\end{bmatrix}-\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}3\\2\end{bmatrix}

The resulting vector is identical to the one drawn at the origin. Translating an arrow does not change its displacement.

Keep this picture.

An arrow and its coordinates are two views of one idea. Place two movements tip to tail and you begin to see vector addition. Stretch a movement and you begin to see scalar multiplication. Those are the next ideas this foundation makes possible.

Intuition

Three groups of people mean three different things by the word vector: an arrow with a length and a direction, an ordered list of numbers, or anything at all that can be added and scaled. The subject lives in the translation between them.

A temperature can be given in Celsius or in Fahrenheit. Neither number is the temperature; each is a way of naming it, and knowing how to move between them is worth more than either alone.

The arrow comes first; the two numbers are read off it afterwards. Three across, then two up — that walk is what $\begin{bmatrix}3\\2\end{bmatrix}$ records.

Same length, same direction, so to a physicist these are one vector. We still draw every arrow from the origin, because that is what gives each one a single pair of numbers.

Walk along $\mathbf{v}$ , then along $\mathbf{w}$ from where you stopped. The single arrow reaching the same place is their sum, and its numbers are the two walks added position by position.

Three descriptions of one thing

The physicist says a vector is an arrow: what fixes it is a length and a direction, and sliding it about changes nothing. The computer scientist says a vector is an ordered list of numbers, and its dimension is how long the list is. The mathematician keeps only what both have in common — that two vectors can be added and that a vector can be scaled by a number — and asks nothing else. The whole of this course is the translation between the first two, with the third waiting at the end.

\mathbf{v}=\begin{bmatrix}3\\2\end{bmatrix}\ \longleftrightarrow\ \text{3 across, 2 up}

Reading the correspondence

Every pair of numbers names exactly one arrow from the origin, and every such arrow names exactly one pair. Nothing is lost either way.
Coordinates are written in a column with square brackets, which is how a vector is told apart from a point written $(3,2)$ .
The order is part of the data: $\begin{bmatrix}3\\2\end{bmatrix}$ and $\begin{bmatrix}2\\3\end{bmatrix}$ are different arrows.
In three dimensions a third axis is added and a vector is a triple, with the same one-to-one correspondence.
The mathematician's version is deliberately vague here. It is made precise by a definite list of eight rules, which this course reaches later; saying so now is better than pretending the phrase 'anything that can be added and scaled' is already a definition.

A negative first coordinate walks left, a negative second walks down. Here $\mathbf{a}$ is two left and three up, and $\mathbf{b}$ is three left and one down.

Scaling stretches, squashes, and with a negative number turns the arrow round. The direction it lies along never changes, which is the whole of what a scalar does.

Applications

Practice

The order is the data

\begin{bmatrix}1\\4\end{bmatrix}\neq\begin{bmatrix}4\\1\end{bmatrix}

One is one across and four up; the other is four across and one up.

Try it

Which arrow does $\begin{bmatrix}-2\\1\end{bmatrix}$ describe?